Problem: How many of the divisors of $8!$ are larger than $7!$?
Explanation: Suppose that $d$ divides $8!$ and that $d>7!$.  Taking the reciprocal of both sides of $d>7!$ and multiplying by $8!$, we find $\frac{8!}{d}<\frac{8!}{7!}=8$.  There are 7 positive integers less than 8, and $d$ can be chosen so that $\frac{8!}{d}$ takes on any of these values, since $\frac{8!}{d}$ ranges over all the divisors of 8! as $d$ ranges over the divisors of $8!$.  Therefore, $\boxed{7}$ divisors of 8! are greater than $7!$.